The Random House Book of Science Fiction Stories has your generic title ("SCIENCE FICTION STORIES" in big letters, "The Random House Book of" in small letters), it has a bunch of spaceships (well three of them) on the cover, and it was published in June 1997, about the very end of the mid 90s. It was published a year earlier in the UK, but with a different title (Space Stories) and a different cover (no spaceships). It seems to contain one of the two stories you described. I'm only mentioning this book on the off chance that you may be mixing up one of the stories with another story you read somewhere else.
The story about people from Earth running up against invisible aliens on a foreign planet (mentioned in your comment) seems to be here: "Scrutiny" by William F. Temple. The spacefarers from Earth are not colonists, they're explorers searching for intelligent alien life. Otherwise the story seems to fit:
While speaking, he was gazing out through a port—and the sudden and inexplicable happened: the outer door of the Module's airlock swung open. Yet there was nobody outside—nor inside—the airlock.
Bruce had also seen it. "That just can't happen," he said, also on the radio set.
"What can't? What's going on there?" the Captain called.
It was Leo who answered. "Maybe someone has come to meet us. If so, he's an invisible man. Could be all the townsfolk are invisible."
Next moment, the inner door of the airlock fell inward on them. They skipped back and it clanged on the floor. The pins of its hinges, which had somehow pulled out, rattled beside it. The denser air in the cabin swooshed out into the planetary atmosphere. Only their suits saved them from near-suffocation at best, death at worst.
[. . .]
The holding screws in the panel of the manual guidance console were spinning anticlockwise. They came clean out and floated loosely, as though weightless. Then the panel itself lifted away, trailing wires and transistors, which began to separate and become detached. It was as though a gang of expert but invisible troubleshooters had set to work to dismantle the apparatus. The thin atmosphere became thick with floating components, as though the Module were in a state of free fall. But it was still resting on the sandy soil and the two men were still captives of gravity.
But the Module seemed to be starting, piecemeal, an independent struggle against that gravity.
Floor bolts began popping like champagne corks. One section of the floor flapped open like a trapdoor.
Bruce exclaimed: "My God, they are invisible! They're wrecking us. Let's get out—quick."
You might take a look at the table of contents to see if any of the other titles ring any bells.
Update. The original poster, Rafael, has found that the story described in the original question is "The Dead Planet" by Edmond Hamilton (first published in Startling Stories, Spring 1946, available at the Internet Archive), also in the The Random House Book of Science Fiction Stories among other places, and online at Google Docs. Here is the ending of Hamilton's story:
We looked at each other, we three tall bird-men of Rigel, as Dril handed us the glasses of pink sanqua. On Tharn's beaked face, in his green eyes, was an expression that told me we were all thinking of the same thing.
He raised the glass that he held in his talons.
"To that great dead race to whom our galaxy owes all," he said. "We will drink to their world by their own name for it. We will drink to Earth."